Firefly algorithm (FA) is a new meta-heuristic optimisation algorithm, which simulates the social behaviour of fireflies based on the flashing and attraction characteristics of fireflies. The standard FA employs a fully attracted model, in which each firefly is attracted by any other brighter firefly in the swarm. However, the fully attracted model may result in slow convergence rate because of too many attractions. In this paper, we propose a new firefly algorithm called FA with random attraction (RaFA), which employs a randomly attracted model. In RaFA, each firefly is attracted by another randomly selected firefly. In order to enhance the global search ability of FA, a concept of Cauchy jump is utilised. Experiments are conducted on a set of well-known benchmark functions. Simulation results show that RaFA outperforms the standard FA and two other improved FAs in terms of solution accuracy and robustness. Compared to the standard FA, RaFA has lower computational time complexity.